"Hydrogen" is the element with atomic number 1 (1 proton). Its common isotope is H-1, (One proton in the nucleus). The other isotopes are deuterium (H-2; one proton and one neutron) and tritium (H-3; one proton and two neutrons).

H2 in a chemical equation means that two hydrogen atoms are bound together.

I'd like to point out that though your thoughts weren't correct, it is possible to separate Hydrogen's heavier isotopes by electrolysis. Water is not only made of H2O, but also other molecules made of different isotopes, such as D2O. Electrolysis "picks out" the lighter isotopes, transforming them to hydrogen gas and oxygen more easily, leaving a higher than normal concentration of heavier isotopes. It seems that it is possible to obtain water made more than 90% out of deuterium monoxide ("heavy water"). Maybe the same applies the separation of T2O from water, in order to make a further electrolysis to finally obtain T2

Tritium doesn't exist in nature. (except in very small quantities, or 'traces' produced by the cosmic rays)

It can be produced in nuclear reactors. Look on wiki: http://en.wikipedia.org/wiki/Tritium. There you will find the reaction of a neutron flux with lithium. This is how Tritium is prepared for the H-WMD.

It is also planned to produce Tritium in fusion reactors. These will be equipped with a blanket of litium where the fusion 14MeV neutrons will produce Tritium. Here tritium will not be used for H-WMDs but as fuel for the fusion reactor itself.